Antagonistic game (Zero-sum game)

Game theory mathematically models conflict situations and offers decision patterns that could be used under different assumptions. The simplest game-theoretic model is a zero-sum game. It is applicable to the conflicts between several parties such that gains of one party are the losses of some other. It could be a rock-paper-scissors game between two players, where for one player to win, the other should lose. Similar examples would be the Odds and Evens game or the matching pennies game. In this topic, for the sake of simplicity, we will focus on the games of two players.

An example of a zero-sum game

Before giving a general definition of a zero-sum game, let's go through an example.

Let two players (the first and the second) play rock-paper-scissors. Any player could take three actions: Rock (R), Paper (P), and Scissors (S). Rock is stronger than scissors, scissors are better than paper and paper is superior to rock. The players choose their moves in advance and play them simultaneously. The player, who chose the stronger move wins while the other loses. If the players choose identical moves, then it is a draw.

Actions, which players can choose, are called strategies, and all the possible strategies of a given player is called the set of strategies of that player. We will denote the set of strategies of the first player by $X$and the set of strategies of the second player by $Y$. For rock-paper-scissors $X=\{R,P,S\}$ and $Y=\{R,P,S\}$.

Let's quantify win and loss. Let the winner get one point and the loser lose one point. In case of a draw, let both players get zero points. Now we can rewrite the algorithm of determining the result of the game for the first player as a function $K(x, y)$. Here $x$ is a strategy of the first player and $y$ is a strategy of the second player.

$K(x, y)= \begin{cases} 0, & \text{if }x=y;\\ 1, & \text{if } (x=R,\,y=S)\text{ or }(x=P,\,y=R)\text{ or }(x=S,\, y=P);\\ -1, & \text{if } (x=R,\,y=P)\text{ or }(x=P,\,y=S)\text{ or }(x=S,\, y=R). \end{cases}$

As payoffs of the second player are equal exactly to $-K(x, y)$, there is no need to explicitly specify payoffs of the second player. So, $K(x,y)$ actually contains information about the payoffs of both players and is therefore called the payoff function of the game.

$X$ and $Y$ are finite sets, so it is convenient and clear to represent $K$ in the form of a matrix. Let rows be strategies of the first player, columns be strategies of the second player and entries be payoffs of the first player.

$$\begin {array} {c c r} & \begin{array}{c} \text{Strategies} \\ \text{of the second player} \end{array}\\ \begin{array}{c} \text{Strategies} \\ \text{of the first player} \end{array}& \begin{array}{|c | c | c | c|} \hline & \text{ R } & \text{ P } & \text{ S } \\\hline R\quad & 0 & -1 & 1 \\\hline P\quad & 1 & 0 & -1 \\\hline S\quad & -1 & 1 & 0 \\\hline \end{array} & \begin{array}{c} \text{Payoffs} \\ \text{of the first player} \end{array} \end{array}$$

General definitions

Definition 1

A zero-sum game in normal form is a tuple $G=(X,Y,K)$ where:

$X, Y$ are non-empty sets which are called the sets of strategies of the first and second players, respectively,

and function $K:X \times Y \to \mathbb{R}$ is called the payoff function of the game. This function maps the set of all possible combinations of elements of sets $X, Y$ to the real numbers.

Usually, the values of $K$ are interpreted as payoffs of the first player. Then it is assumed that the payoffs of the second player are equal to $-K$. That property is the essence of zero-sum games. It represents the assumption that the gains of the first player are the losses of the second one and vice versa so the total sum of their payoffs always remains zero.

If sets $X$ and $Y$ are finite then it is easy to represent the payoff function $K$ as a matrix. We have done that already to the rock-paper-scissors game.

Definition 2

If $G=(X, Y, K)$ is a zero-sum game and $X, Y$ are finite sets then $G$ is called a matrix game.

Conclusion

In this topic, we explored the concepts of a zero-sum game and a matrix game. We learned how to build a mathematical model of a zero-sum game and discussed the assumptions of this model. Here are the main points that we have learned today:

• In a zero-sum game if we sum both players' payoffs we will get 0, thus we can say that if payoffs of one player are represented by $K$, the payoffs of the second player will be represented by $-K$
• All actions that a player can take are contained in the set of strategies of the corresponding player
• The payoff function of the game maps contains all the information about the payoffs of both players
• Usually, the matrix is the best way to represent the game and its outcomes for two players
• Whether the game is played simultaneously or not changes the whole scenario drastically, as in the example of rock-paper-scissors it wouldn't work